A Theil-type estimate in multiple linear regression and developing a new BIC for detecting change-points

Gang Shen, Purdue University

Abstract

The Theil estimate for simple linear regression coefficients has been around for a long time and has many important applications, however it has not been considered for multiple linear regression in literature, much less its asymptotics. Busarova et al [11] proposed a Theil-type estimate for multiple linear regression via Oja’s median and derived its asymptotics for the case of iid random covariates, whereas the case of deterministic covariates, which is more interesting case in application, is still untouched yet. Based on the convexity lemma, Shen [51] showed the asymptotics of this Theil-type estimate in the case of deterministic covariates. An example of orthogonal design was also given there as to show its relatively high asymptotic efficiency. More important, the approach is quite simple and may be easily extended to the M-estimate problem of many convex processes, including convex U-processes. As one more illustration of this powerful approach, we also showed asymptotics of Oja’s median. Inference about the time of change in parameters in a model, namely, a so-called change-point, has been the subject of considerable interest. Raftery [40] proposed a Bayesian approach on the change-point problem via Bayes Factors for tests about the null hypothesis of no change-point against an alternative of a change-point having occurred in the given data in the time interval under consideration. We replaced the above test for a change-point and associated estimates with a simpler Bayesian approach through construction of an approximation like Schwartz’s BIC which doesn’t require specifying a prior. As expected, our new approximation lBIC leads to a penalty that is different from the standard Schwartz type penalty. However, like Schwartz BIC, our new approximation has an error up to the order of O(1) and consistent under both null and alternative hypothesis. The methodology, but not the proof, has a straightforward extension to much more complex problems. We apply the new method to simulated as well as real data with satisfactory results.

Degree

Ph.D.

Advisors

Ghosh, Purdue University.

Subject Area

Statistics

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